3D printing "be like the 3D view"
The object is non-manifold and the main reason why are these edges holding the top part because technically they aren't holding it:



For now printer won't be able to decide where is the inner part of that, where is outer, just as Blender can't calculate normals for the object correctly.

If you really want to print this object the easiest way to work precisely is to duplicate and separate to another object the outline for those curved extruded parts in order to cut them with Knife project / Boolean once they are part of another object.

First copy the top part of the object with which is the profile of the curved part extruded from the bottom. Then either hide those vertices with Mask modifier or delete them. Then dissolve the vertices corresponding to the ones removed from the top, - they are just going through the bottom part faces and are creating intersecting geometry. 



In the end you want to get the bottom part only without any additional loopcuts:



From here on get duplicated curved profile of the deleted geometry, extrude it to get a line of faces in the top and line of faces going down:



Add Boolean modifier to the original object shown above, set it to Difference and use this profile as cutter. Below the cutter is hidden and new geometry create with difference is shown:



Do the same for the other part. Then either extrude top part once more or (as is done in the file below) copy the one from the original mesh, clean it up from doubles and unnecessary edges to get even quads and join meshes (Boolean has to be applied first). Then select both lines of faces and run Bridge Edge Loops.



Here you can convert tris to quads or not - it's not quite important for printer, the geometry here is more or less clean. 
What matters for printer here are quads / ngons which might not be flat (in your model this is the case), as well as overhanging faces which might or might not require fixing though this is more printing related question than Blender.

Download the file